Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats


9    MEASURABLE FUNCTIONS       135

The sets obtained from the open (or closed) sets of X by iterated application
of the operations (i) and (ii) above are therefore also universally measurable.
The definition of a ^-measurable set A also shows that there exists a uni-
versally measurable set B contained in A such that A  B is /^-negligible. By
applying this result to X  A, we see that there also exists a universally
measurable set C containing A such that C  A is /(-negligible.

A mapping u of X into a topological space Y is said to be measurable
(with respect to ju) or ^-measurable if there exists a partition of X into a
sequence of compact sets Kn and a ^-negligible set N such that each of the
restrictions u \ Kn is continuous. The mapping u is said to be universally
measurable if it is measurable for all measures on X. A continuous function
is universally measurable,

For a subset A of X to be measurable (resp. universally measurable) it is
necessary and sufficient that its characteristic function (pA should be measur-
able (resp. universally measurable). For if we have a partition of X consisting
of a sequence of compact sets Kn and a negligible set N, such that <pA | Kn is
continuous for all n, then Kn is the union of two disjoint compact subsets,
namely A n Kn and KM - (A n KJ ((3.11.4) and (3.17.3)), and A is the union
of the sets A n Kn and A n N, and is therefore measurable. Conversely, if
A is measurable, then so is X  A (13.9.3), hence there exists a partition of
X consisting of a sequence of compact sets Kn and a negligible set N, such that
either KM c A or Kn c X  A for all n. Consequently <pA | Kn is continuous
for all n.

(1 3.9.4)   Let u be a mapping ofX. into a topological space Y. Then the following
three properties are equivalent:

(a)    u is ^.-measurable.

(b)    For each compact subset K of X and each s > 0, there exists a com-
pact subset K' of K such that ^(K  K') ^  and such that the restriction of
u to K' is continuous.

(c)    For each compact subset K of X, there exists a partition ofK. consisting
of a sequence (LM) of compact sets and a ^-negligible set M, such that each of
the restrictions u \ Ln is continuous.

To show that (a) implies (c), we remark that the hypothesis (a) implies
the existence of a partition of X consisting of a sequence (KJ of compact
sets and a negligible set N, such that u \ Kn is continuous for all n ; hence (c)
is satisfied by taking LM = K n Kn and M = K n N.

To show that (c) implies (b), we remark that it follows from (c) that

(13.8.7), and hence that there is an integer n such that

n-l

/i(K  K') g e, where K' = (J Lt . Clearly K' satisfies the conditions of (b).serve that there exists a se-